Problem: Let $g$ be a twice differentiable function, and let $g(-6)=-1$, $g'(-6)=0$, and $g''(-6)=-3$. What occurs in the graph of $g$ at the point $(-6,-1)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6,-1)$ is a minimum point. (Choice B) B $(-6,-1)$ is a maximum point. (Choice C) C There's not enough information to tell.
Answer: Since $g'(-6)=0$, we know that $x=-6$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $g$ at this point according to these three cases: If $g''(-6)>0$, the graph of $g$ has a minimum point at $x=-6$. If $g''(-6)<0$, the graph of $g$ has a maximum point at $x=-6$. If $g''(-6)=0$, the test is inconclusive. [Why is this so?] We are given that $g''(-6)=-3<0$. Therefore, $(-6,-1)$ is a maximum point.